We present the Virgo Environment Traced in CO (VERTICO) survey, a new effort to map 12 CO(2 − 1), 13 CO(2−1), and C 18 O(2−1) in 51 Virgo Cluster galaxies with the Atacama Compact Array, part of the Atacama Large Millimeter/submillimeter Array (ALMA). The primary motivation of VERTICO is to understand the physical mechanisms that perturb molecular gas disks, and therefore star formation and galaxy evolution, in dense environments. This first paper contains an overview of VERTICO's design and sample selection, 12 CO(2−1) observations, and data reduction procedures. We characterize global 12 CO(2 − 1) fluxes and molecular gas masses for the 49 detected VERTICO galaxies, provide upper limits for the two non-detections, and produce resolved 12 CO(2 − 1) data products (median